Working with metric graphs

Introduction

Networks such as street or river networks are examples of metric graphs. A compact metric graph $\Gamma$ consists of a set of finitely many vertices $\mathcal{V}=\{v_i\}$ and a finite set $\mathcal{E}=\{e_j\}$ of edges connecting the vertices. Each edge $e$ is a curve of finite length $l_e$ that connects two vertices. These curves are parameterized by arc length and a location $s\in \Gamma$ is a position on an edge, and can thus be represented as a touple $(e,t)$ where $t\in[0,l_e]$ . Compared to regular graphs, where one typically defines functions on the vertices, we are for metric graphs interested in function that are defined on both the vertices and the edges.

In this vignette we will introduce the metric_graph class of the MetricGraph package. This class provides a user friendly representation of metric graphs, and we will show how to use the class to construct and visualize metric graphs, add data to them, and work with functions defined on the graphs.

For details about Gaussian processes and inference on metric graphs, we refer to the Vignettes

Constructing metric graphs

Basic constructions and properties

A metric graph can be constructed in two ways. The first is to specify all edges in the graph as a list object, where each entry is a matrix. To illustrate this, we first construct the following edges

edge1 <- rbind(c(0,0),c(1,0))
edge2 <- rbind(c(0,0),c(0,1))
edge3 <- rbind(c(0,1),c(-1,1))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
edge4 <- cbind(sin(theta),1+ cos(theta))
edges = list(edge1, edge2, edge3, edge4)

We can now create the graph based on the edges object as follows

graph <- metric_graph$new(edges = edges)
graph$plot()

The plot function that was used to create the plot above has various parameters to set the sizes and colors of the vertices and edges, and it has a plotly argument to visualize the graph in 3D. For this to work, the plotly library must be installed.

graph$plot(plotly = TRUE, vertex_size = 5, vertex_color = "blue",
           edge_color = "red", edge_width = 2)

## Warning: The `plotly` argument of `plot()` is deprecated as of MetricGraph 1.3.0.9000.
## ℹ Please use the `type` argument instead.
## ℹ The argument `plotly` was deprecated in favor of the argument `type`.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

It is also important to know that the 2d version of the plot() method returns a ggplot2 object and can be modified as such. For instance:

p <- graph$plot()
p + ggplot2::labs(title = "Metric graph",
          x = "long", y = "lat")

Similarly, the 3d version of the plot() method returns a plotly object that can also be modified. For instance:

p <- graph$plot(plotly = TRUE)
p <- plotly::layout(p, title = "Metric graph", 
              scene = list(xaxis=
              list(title = "Long"),yaxis=list(title = "Lat")))
p

We can now obtain various properties of the graph: The vertex matrix, which specifies the Euclidian coordinates of the vertices is

graph$V

##       X Y
## [1,]  0 0
## [2,]  1 0
## [3,]  0 1
## [4,] -1 1

and the edge matrix that specified the edges of the graph (i.e., which vertices that are connected by edges) is

graph$E

##      [,1] [,2]
## [1,]    1    2
## [2,]    1    3
## [3,]    3    4
## [4,]    1    4

To obtain the geodesic (shortest path) distance between the vertices, we can use the function compute_geodist:

graph$compute_geodist()
graph$geo_dist

## $.vertices
##          [,1]     [,2] [,3]     [,4]
## [1,] 0.000000 1.000000    1 1.570729
## [2,] 1.000000 0.000000    2 2.570729
## [3,] 1.000000 2.000000    0 1.000000
## [4,] 1.570729 2.570729    1 0.000000

The second option it to construct the graph using two matrices V and E that specify the locations (in Euclidean space) of the vertices and the edges. In this case, it is assumed that the graph only has straight edges:

V <- rbind(c(0, 0), c(1, 0), c(0, 1), c(-1, 1))
E <- rbind(c(1, 2), c(1, 3), c(3, 4), c(4, 1))
graph2 <- metric_graph$new(V = V, E = E)

## Starting graph creation...

## LongLat is set to FALSE

## Setting up edges

## Merging close vertices

## Total construction time: 0.19 secs

graph2$plot()

A third option is to create a graph from a SpatialLines object:

library(sp)
line1 <- Line(rbind(c(0,0),c(1,0)))
line2 <- Line(rbind(c(0,0),c(0,1)))
line3 <- Line(rbind(c(0,1),c(-1,1)))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
line4 <- Line(cbind(sin(theta),1+ cos(theta)))
Lines = sp::SpatialLines(list(Lines(list(line1),ID="1"),
                              Lines(list(line2),ID="2"),
                              Lines(list(line3),ID="3"),
                              Lines(list(line4),ID="4")))

graph <- metric_graph$new(edges = Lines)                    
graph$plot()

A final option is to create from a MULTILINESTRING object:

library(sf)
line1 <- st_linestring(rbind(c(0,0),c(1,0)))
line2 <- st_linestring(rbind(c(0,0),c(0,1)))
line3 <- st_linestring(rbind(c(0,1),c(-1,1)))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
line4 <- st_linestring(cbind(sin(theta),1+ cos(theta)))
multilinestring = st_multilinestring(list(line1, line2, line3, line4))

graph <- metric_graph$new(edges = multilinestring)                    
graph$plot()

Tolerances for merging vertices and edges

The constructor of the graph has one argument, perform_merges and an additional argument tolerance which is used for connecting edges that are close in Euclidean space. Specifically, the tolerance argument is given as a list with three elements:

vertex_vertex vertices that are closer than this number are merged (the default value is 1e-7)
vertex_edge if a vertex at the end of one edge is closer than this number to another edge, this vertex is connected to that edge (the default value is 1e-7)
edge_edge if two edges at some point are closer than this number, a new vertex is added at that point and the two edges are connected (the default value is 0 which means that the option is not used)

These options are often needed when constructing graphs based on real data, for example from OpenStreetMap as we will see later. To illustrate these options, suppose that we want to construct a graph from the following three edges. We will set perform_merges to TRUE, to show how they are merged with the default choice of tolerances.

edge1 <- rbind(c(0,0),c(1,0))
edge2 <- rbind(c(0,0.03),c(0,0.75))
edge3 <- rbind(c(-1,1),c(0.5,0.03))
edge4 <- rbind(c(0,0.75), c(0,1))
edge5 <- rbind(c(-1,0), c(-1, 0.9995))
edges =  list(edge1, edge2, edge3, edge4, edge5)
graph3 <- metric_graph$new(edges = edges, perform_merges = TRUE)
graph3$plot(degree = TRUE)
print(graph3$nV)

## [1] 8

We added the option degree=TRUE to the plot here to visualize the degrees of each vertex. As expected, one sees that all vertices have degree 1, and none of the three edges are connected. If these are streets in a street network, one might suspect that the two vertices at $(0,0)$ and $(0,0.03)$ really should be the same vertex so that the two edges are connected. This can be adjusted by increasing the vertex_vertex tolerance:

graph3 <- metric_graph$new(edges = edges, perform_merges = TRUE,
                             tolerance = list(vertex_vertex = 0.05))
graph3$plot(degree = TRUE)

One might also want to add the vertex at $(0.5, 0.03)$ as a vertex on the first edge, so that the two edges there are connected. This can be done by adjusting the vertex_edge tolerance:

graph3 <- metric_graph$new(edges = edges, perform_merges = TRUE,
                                       tolerance = list(vertex_vertex = 0.05,
                                                           vertex_edge = 0.1))
graph3$plot(degree = TRUE)

We can see that the vertex at $(0.5,0)$ was indeed connected with the edge from $(0,0)$ to $(1,0)$ and that vertex now has degree 3 since it is connected with three edges. One can also note that the edges object that was used to create the graph is modified internally in the metric_graph object so that the connections are visualized correctly in the sense that all edges are actually shown as connected edges.

Finally, to add a vertex at the intersection between edge2 and edge3 we can adjust the edge_edge tolerance:

graph3 <- metric_graph$new(edges = edges, perform_merges = TRUE,
                                        tolerance = list(vertex_vertex = 0.2,
                                                           vertex_edge = 0.1,
                                                           edge_edge = 0.001))
graph3$plot(degree = TRUE)

Now, the structure of the metric graph does not change if we add or remove vertices of degree 2. Because of this,one might want to remove vertices of degree 2 since this can reduce computational costs. This can be done by setting the remove_deg2 argument while creating the graph:

graph3 <- metric_graph$new(edges = edges, perform_merges = TRUE,
                                        tolerance = list(vertex_vertex = 0.2,
                                                           vertex_edge = 0.1,
                                                           edge_edge = 0.001),
                           remove_deg2 = TRUE)
graph3$plot(degree = TRUE)

Observe that one vertex of degree 2 was not removed. This is due to the fact that if we consider the direction, this vertex has directions incompatible for removal. We can see this by setting the argument direction=TRUE in the plot() method:

graph3$plot(direction = TRUE)

We can also avoid performing these merges by setting using the default constructor (which is equivalent to set the perform_merges argument to FALSE):

graph4 <- metric_graph$new(edges = edges)
graph4$plot(degree = TRUE)

Observe that the two vertices around the point $(-1,1)$ were merged. This is because there is an additional step to merge close vertices that is performed even with perform_merges set to FALSE. To also skip this merge, we can also set merge_close_vertices to FALSE:

graph4 <- metric_graph$new(edges = edges, merge_close_vertices=FALSE)
graph4$plot(degree = TRUE)

We can see now that the two vertices around the point $(-1,1)$ were not merged.

Edge lengths

Whenever we create a metric graph, the edge lengths will be automatically computed by obtaining the lengths of the edges. However, sometimes one might want to provide manual edge lengths.

Let us consider the following example:

library(sp)
line1 <- Line(rbind(c(0,0),c(1,0)))
line2 <- Line(rbind(c(0,0),c(0,1)))
line3 <- Line(rbind(c(0,1),c(-1,1)))
theta <- seq(from=pi,to=3*pi/2,length.out = 25)
line4 <- Line(cbind(sin(theta),1+ cos(theta)))
Lines = sp::SpatialLines(list(Lines(list(line1),ID="1"),
                              Lines(list(line2),ID="2"),
                              Lines(list(line3),ID="3"),
                              Lines(list(line4),ID="4")))

graph <- metric_graph$new(edges = Lines)                 
graph$plot()

We can obtain the edge lenghts by using the get_edge_lengths() method:

graph$get_edge_lengths()

## [1] 1.000000 1.000000 1.000000 1.570516

Observer that we know that the actual length of edge 4 is $\pi/2$ , which is not quite right (observe that edge 4 is an approximation of a quarter of a circle by a polygon with 25 points):

pi/2 - graph$get_edge_lengths()[4]

## [1] 0.0002803513

We can provide the actual edge lengths, by using the set_manual_edge_lengths() method:

graph$set_manual_edge_lengths(c(1,1,1,pi/2))
graph$get_edge_lengths()

## [1] 1.000000 1.000000 1.000000 1.570796

We can also manually provide the edge lengths when creating the metric graph:

graph <- metric_graph$new(edges = Lines, manual_edge_lengths = c(1,1,1,pi/2))
graph$get_edge_lengths()

## [1] 1.000000 1.000000 1.000000 1.570796

However, it is important to note that if one passes the manual_edge_lengths when creating the graph, it will automatically set perform_merges to FALSE, as there is no consistent way to obtain the edge lengths after the merges from the manual edge lengths. Nevertheless, merge_close_vertices can be set to TRUE.

Characteristics of the graph

A brief summary of the main characteristics of the graph can be obtained by the compute_characteristics() method:

graph3$compute_characteristics()

We can, then, view those characteristics by simply printing the metric_graph object:

graph3

## A metric graph with  6  vertices and  6  edges.
## 
## Vertices:
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Lengths: 
##       Min: 0.3533333  ; Max: 2.190873  ; Total: 4.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: unknown

## To check if the graph satisfies the distance consistency, run the `check_distance_consistency()` method.

##   Has Euclidean edges: unknown

## To check if the graph has Euclidean edges, run the `check_euclidean()` method.

Observe that we do not know that the graph has Euclidean edges. We can check by using the check_euclidean() method:

graph3$check_euclidean()

Let us view the characteristics again:

graph3

## A metric graph with  6  vertices and  6  edges.
## 
## Vertices:
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Lengths: 
##       Min: 0.3533333  ; Max: 2.190873  ; Total: 4.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE

It can happen that we have that the graph does not have Euclidean edges without the need to check the distance consistency. However, if one wants to check if the graph satisfies the distance consistency assumption, one can run the check_distance_consistency() function:

graph3$check_distance_consistency()

Further, the individual characteristics can be accessed (after running the compute_characteristics() method) through the characteristics component of the graph object:

graph3$characteristics

## $has_loops
## [1] FALSE
## 
## $connected
## [1] TRUE
## 
## $has_multiple_edges
## [1] FALSE
## 
## $is_tree
## [1] FALSE
## 
## $distance_consistency
## [1] TRUE
## 
## $euclidean
## [1] TRUE

Summaries of metric graphs

We can also obtain a summary of the informations contained on a metric graph object by using the summary() method. Let us obtain a summary of informations of graph3:

summary(graph3)

## A metric graph object with:
## 
## Vertices:
##   Total: 6 
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 6 
##   Lengths: 
##       Min: 0.3533333  ; Max: 2.190873  ; Total: 4.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: The graph has no mesh! 
## 
## Data: The graph has no data!
## 
## Tolerances: 
##   vertex-vertex:  0.2 
##   vertex-edge:  0.1 
##   edge-edge:  0.001

Observe that there are some quantities that were not computed on the summary. We can see how to compute them by setting the argument messages to TRUE:

summary(graph3, messages = TRUE)

## A metric graph object with:
## 
## Vertices:
##   Total: 6 
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 6 
##   Lengths: 
##       Min: 0.3533333  ; Max: 2.190873  ; Total: 4.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE

## To compute the Laplacian, run the 'compute_laplacian()' method.

##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE

## To compute the resistance distances, run the 'compute_resdist()' method.

## To compute the finite element matrices, run the 'compute_fem()' method.

## 
## Mesh: The graph has no mesh!

## To build the mesh, run the 'build_mesh()' method.

## 
## Data: The graph has no data!

## To add observations, use the 'add_observations()' method.

## 
## Tolerances: 
##   vertex-vertex:  0.2 
##   vertex-edge:  0.1 
##   edge-edge:  0.001

Finally, the summary() can also be accessed from the metric graph object:

graph3$summary()

## A metric graph object with:
## 
## Vertices:
##   Total: 6 
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 6 
##   Lengths: 
##       Min: 0.3533333  ; Max: 2.190873  ; Total: 4.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: The graph has no mesh! 
## 
## Data: The graph has no data!
## 
## Tolerances: 
##   vertex-vertex:  0.2 
##   vertex-edge:  0.1 
##   edge-edge:  0.001

Vertices and edges

The metric graph object has the edges and vertices elements in its list. We can get a quick summary from them by calling these elements:

graph3$vertices

## Vertices of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Summary: 
##      x         y Degree Indegree Outdegree Problematic
## 1  0.0 0.0000000      2        0         2        TRUE
## 2  1.0 0.0000000      1        1         0       FALSE
## 3  0.5 0.0300000      3        2         1       FALSE
## 4  0.0 1.0000000      1        1         0       FALSE
## 5 -1.0 0.0000000      1        0         1       FALSE
## 6  0.0 0.3533333      4        2         2       FALSE

and

graph3$edges

## [[1]]
## Edge 1 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##    x    y
##  0.0 0.00
##  0.5 0.03
## 
## Coordinates of the edge:
##    x    y
##  0.0 0.00
##  0.5 0.03

## Relative positions of the coordinates on the graph edges were not computed.

## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 2 
## Edge length: 0.5008992 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1 
## 
## 
## [[2]]
## Edge 2 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##  x         y
##  0 0.0000000
##  0 0.3533333
## 
## Coordinates of the edge:
##  x         y
##  0 0.0000000
##  0 0.3533333

## Relative positions of the coordinates on the graph edges were not computed.
## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 2 
## Edge length: 0.3533333 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1 
## 
## 
## [[3]]
## Edge 3 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##   x         y
##  -1 0.0000000
##   0 0.3533333
## 
## Coordinates of the edge:
##   x         y
##  -1 0.0000000
##  -1 1.0000000
##  -1 1.0000000
##   0 0.3533333

## Relative positions of the coordinates on the graph edges were not computed.
## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 4 
## Edge length: 2.190873 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1 
## 
## 
## [[4]]
## Edge 4 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##  x         y
##  0 0.3533333
##  0 1.0000000
## 
## Coordinates of the edge:
##  x         y
##  0 0.3533333
##  0 0.7500000
##  0 0.7500000
##  0 1.0000000

## Relative positions of the coordinates on the graph edges were not computed.
## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 4 
## Edge length: 0.6466667 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1 
## 
## 
## [[5]]
## Edge 5 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##    x         y
##  0.0 0.3533333
##  0.5 0.0300000
## 
## Coordinates of the edge:
##    x         y
##  0.0 0.3533333
##  0.5 0.0300000

## Relative positions of the coordinates on the graph edges were not computed.
## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 2 
## Edge length: 0.5954363 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1 
## 
## 
## [[6]]
## Edge 6 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##    x    y
##  0.5 0.03
##  1.0 0.00
## 
## Coordinates of the edge:
##    x    y
##  0.5 0.03
##  1.0 0.00

## Relative positions of the coordinates on the graph edges were not computed.
## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 2 
## Edge length: 0.5008992 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1

We can also look at individual edges and vertices. For example, let us look at vertice number 2:

graph3$vertices[[2]]

## Vertex 2 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Summary: 
##  x y Degree Indegree Outdegree Problematic
##  1 0      1        1         0       FALSE

Similarly, let us look at edge number 4:

graph3$edges[[4]]

## Edge 4 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##  x         y
##  0 0.3533333
##  0 1.0000000
## 
## Coordinates of the edge:
##  x         y
##  0 0.3533333
##  0 0.7500000
##  0 0.7500000
##  0 1.0000000

## Relative positions of the coordinates on the graph edges were not computed.

## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 4 
## Edge length: 0.6466667 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1

We have a message informing us that the relative positions of the edges were not computed. They are used to produce better plots when using the plot_function() method, see the section “Improving the plot obtained by plot_function” below. To compute the relative positions, we run the compute_PtE_edges() method:

graph3$compute_PtE_edges()

Now, we take a look at edge number 4 again:

graph3$edges[[4]]

## Edge 4 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##  x         y
##  0 0.3533333
##  0 1.0000000
## 
## Coordinates of the edge:
##  x         y
##  0 0.3533333
##  0 0.7500000
##  0 0.7500000
##  0 1.0000000
## 
## Relative positions of the edge:
##  Edge number Distance on edge
##            2        1.0000000
##            4        0.6134021
##            4        0.6134021
##            4        1.0000000
## 
## Total number of coordinates: 4 
## Edge length: 0.6466667 
## Weight: 1 
## Kirchhoff weight: 1 
## 
## Directional weight: 1

Understanding coordinates on graphs

The locations of the vertices are specified in Euclidean coordinates. However, when specifying a position on the graph, it is not practical to work with Euclidean coordinates since not all locations in Euclidean space are locations on the graph. It is instead better to specify a location on the graph by the touple $(i, t)$ , where $i$ denotes the number of the edge and $t$ is the location on the edge. The location $t$ can either be specified as the distance from the start of the edge (and then takes values between 0 and the length of the edge) or as the normalized distance from the start of the edge (and then takes values between 0 and 1). The function coordinates can be used to convert between coordinates in Euclidean space and locations on the graph. For example the location at distance 0.2 from the start of the second edge is:

graph$coordinates(PtE = matrix(c(2, 0.2), 1,2), normalized = FALSE)

##      [,1] [,2]
## [1,]    0  0.2

In this case, since the edge has length 1, the location of the point at normalized distance 0.2 from the start of the edge is the same:

graph$coordinates(PtE = matrix(c(2, 0.2), 1,2), normalized = TRUE)

##      [,1] [,2]
## [1,]    0  0.2

The function can also be used to find the closest location on the graph for a location in Euclidean space:

graph$coordinates(XY = matrix(c(0, 0.2), 1,2))

##      [,1] [,2]
## [1,]    2  0.2

In this case, the normalized argument decides whether the returned value should be given in normalized distance or not.

Methods for working with real data

To illustrate the useage of metric_graph on some real data, we use the osmdata package to download data from OpenStreetMap. In the following code, we extract highways in the city of Copenhagen:

library(osmdata)
call <- opq(getbb("Copenhagen"))
call <- add_osm_feature(call, key = "highway",value=c("motorway", "primary",
                                                      "secondary"))
data_sp <- osmdata_sp(call)

graph5 <- metric_graph$new(SpatialLines(data_sp$osm_lines@lines))

## Warning in initialize(...): There is at least one edge of length zero. Please,
## consider redefining the graph.

graph5$plot(vertex_size = 0)

There are a few things to note about data like this. The first is that the coordinates are given in Longitude and Latitude. Because of this, the edge lengths are by default given in degrees, which may result in very small numbers:

range(graph5$get_edge_lengths())

## [1] 0.00000000 0.05630008

This may cause numerical instabilities when dealing with random fields on the graph, and it also makes it difficult to interpret results (unless one has a good intuition about distances in degrees). To avoid such problems, it is better to set the longlat argument when constructing the graph:

graph5 <- metric_graph$new(SpatialLines(data_sp$osm_lines@lines), longlat = TRUE)

## Warning in initialize(...): There is at least one edge of length zero. Please,
## consider redefining the graph.

This tells the constructor that the coordinates are given in Longitude and Latitude and that distances should be calculated in km. So if we now look at the edge lengths, they are given in km:

range(graph5$get_edge_lengths())

## [1] 0.00000000 0.05630008

Alternatively, and more conveniently, whenever the SpatialLines object has a valid proj4string, it will automatically use this proj4string to set the correct coordinate reference system (CRS) in the metric graph. Above, we ended up removing the proj4string when we created the SpatialLines object from the resulting osm_lines. Let us, instead, build the graph directly using the osmdata_sp object:

graph5 <- metric_graph$new(data_sp)

## Warning in initialize(...): There is at least one edge of length zero. Please,
## consider redefining the graph.

We can see now that graph5 automatically has the correct CRS:

graph5

## A metric graph with  1458  vertices and  1490  edges.
## 
## Vertices:
##   Degree 1: 48;  Degree 2: 1310;  Degree 3: 88;  Degree 4: 12; 
##   With incompatible directions:  3 
## 
## Edges: 
##   Lengths: 
##       Min: 0  ; Max: 4.05379  ; Total: 326.2565 
##   Weights: 
##       Columns: osm_id name access:lanes alt_name bicycle bicycle:lane bicycle:left bridge bus bus:lanes busway:right change:lanes check_date:cycleway check_date:surface communication covered cutting:right cycleway cycleway:both cycleway:both:lane cycleway:left cycleway:left:lane cycleway:left:oneway cycleway:right cycleway:right:lane cycleway:right:width cycleway:track:width cycleway:width destination destination:int_ref:lanes destination:lanes destination:ref destination:ref:to destination:symbol fixme foot foot:left footway:left hazard highway horse int_ref lane_markings lanes lanes:backward lanes:forward layer lit mapillary maxheight maxheight:signed maxspeed maxspeed:advisory maxspeed:variable maxweight:signed minspeed:lanes motorcar name:etymology name:etymology:wikidata noname note official_ref old_name old_name:wikipedia oneway oneway:bicycle operator operator:en operator:type operator:wikidata operator:wikipedia parking:both parking:left parking:right parking:right:orientation placement placement:backward placement:end placement:forward placement:start ref shoulder shoulder:right sidewalk sidewalk:both:surface sidewalk:left sidewalk:left:surface sidewalk:right sidewalk:right:surface smoothness source source:bicycle source:lanes source:lanes:date source:maxspeed source:official_ref source:start_date start_date surface temporary:date_off temporary:date_on temporary:oneway:bicycle toll tunnel tunnel:alt_name:da tunnel:alt_name:sv tunnel:name tunnel:name:da tunnel:name:de tunnel:name:en tunnel:name:sv tunnel:official_name turn turn:bus:lanes turn:lanes turn:lanes:backward turn:lanes:both_ways turn:lanes:forward unsigned_ref vehicle wikidata wikipedia .weights 
##   That are circles:  1 
## 
## Graph units: 
##   Vertices unit:  degree  ; Lengths unit:  km 
## 
## Longitude and Latitude coordinates:  TRUE
##   Which spatial package:  sp 
##   CRS:  +proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs +towgs84=0,0,0

This also allows us to plot the graph interactively with the underlying world map, if the data has a coordinate reference system, by setting type = "mapview" in the plot() method:

graph5$plot(vertex_size = 0, type = "mapview")

## Loading required namespace: mapview

David Bolin, Alexandre B. Simas, and Jonas Wallin

Created: 2022-11-23. Last modified: 2024-10-21.